Functions of a real variable

Let us call f a function of a real variable, which takes real values. We write f : ℝ → ℝ : x → f(x). The graph of a function f : ℝ → ℝ is the set of points (x, y) ∈ ℝ^{2} so that y is the image of x under f. We denote it by Graph(f). More concisely,

Graph(f) := {(x, f(x)) : x ∈ ℝ and f(x) exists}.

It is important to note that, for all x ∈ ℝ, there is at most one y ∈ ℝ such that (x, y) ∈ Graph(f).

You are expected to be able to draw the graphs of elementary functions, and to perform basic manipulations of graphs (horizontal and vertical translation, symmetry w.r.t. the y-axis, the x-axis and the origin, dilatations and contractions).

Try matching the following functions to the graphs below.

(a) f_{1}(x) = ln x (g) f_{7}(x) = cos 2x + 1

(b) f_{2}(x) = arcsin x (h) f_{8}(x) = x^{2} − 2x

(c) f_{3}(x) = 1/x (i) f_{9}(x) = x^{3}

(d) f_{4}(x) = |3 − x| (j) f_{10}(x) = e^{−x^{2}}

(e) f_{5}(x) = x − |x| (k) f_{11}(x) = |4 − x^{2}|

(f) f_{6}(x) = sin (x + π) (l) f_{12}(x) = tan x

graph 1  graph 2  graph 3

graph 4  graph 5  graph 6

graph 7  graph 8  graph 9

graph 10  graph 11  graph 12

Solutions: f_{1}(x): graph 5; f_{2}(x) : graph 8; f_{3}(x) : graph 6; f_{4}(x) : graph 12; f_{5}(x) : graph 7; f_{6}(x) : graph 4; f_{7}(x) : graph 11; f_{8}(x) : graph 2; f_{9}(x) : graph 9; f_{1}0(x) : graph 3; f_{1}1(x) : graph 1; f_{1}2(x) : graph 10